What’s an equation that could determine the probability that an object with mass will balance on any given side/face with a certain surface area?

Question

You have access to all properties of the object to synthesize this formula.

For example, a pencil that weighs 10g on the face of the point with a minuscule surface area should return close to zero. Whereas a textbook that weighs 75g and is laying on its side, a larger surface area, would return a number closer to 1.

Would I need a constant for comparison between weight and surface area of a face?

(Thinking about this for fun.)

Answer 1

To calculate a probability, some of the variables need to be random.

You could ask, for instance, whether the object will remain balanced if subjected to an impulse drawn from some specified distribution.

To answer such questions you can draw from several tools: 1. The fact that a rigid object is stably at rest on a flat surface if and only if its center of gravity lies over the convex hull of its support; 2. Conservation of energy: you can compute the minimum increase in potential energy require to tilt the object in each direction so that the object is no longer stable, and use that to estimate the above probability.